<HTML>
<BODY BGCOLOR="white">
<PRE>
<FONT color="green">001</FONT>    /*<a name="line.1"></a>
<FONT color="green">002</FONT>     * Licensed to the Apache Software Foundation (ASF) under one or more<a name="line.2"></a>
<FONT color="green">003</FONT>     * contributor license agreements.  See the NOTICE file distributed with<a name="line.3"></a>
<FONT color="green">004</FONT>     * this work for additional information regarding copyright ownership.<a name="line.4"></a>
<FONT color="green">005</FONT>     * The ASF licenses this file to You under the Apache License, Version 2.0<a name="line.5"></a>
<FONT color="green">006</FONT>     * (the "License"); you may not use this file except in compliance with<a name="line.6"></a>
<FONT color="green">007</FONT>     * the License.  You may obtain a copy of the License at<a name="line.7"></a>
<FONT color="green">008</FONT>     *<a name="line.8"></a>
<FONT color="green">009</FONT>     *      http://www.apache.org/licenses/LICENSE-2.0<a name="line.9"></a>
<FONT color="green">010</FONT>     *<a name="line.10"></a>
<FONT color="green">011</FONT>     * Unless required by applicable law or agreed to in writing, software<a name="line.11"></a>
<FONT color="green">012</FONT>     * distributed under the License is distributed on an "AS IS" BASIS,<a name="line.12"></a>
<FONT color="green">013</FONT>     * WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.<a name="line.13"></a>
<FONT color="green">014</FONT>     * See the License for the specific language governing permissions and<a name="line.14"></a>
<FONT color="green">015</FONT>     * limitations under the License.<a name="line.15"></a>
<FONT color="green">016</FONT>     */<a name="line.16"></a>
<FONT color="green">017</FONT>    package org.apache.commons.math3.analysis.polynomials;<a name="line.17"></a>
<FONT color="green">018</FONT>    <a name="line.18"></a>
<FONT color="green">019</FONT>    import java.util.ArrayList;<a name="line.19"></a>
<FONT color="green">020</FONT>    import java.util.HashMap;<a name="line.20"></a>
<FONT color="green">021</FONT>    import java.util.List;<a name="line.21"></a>
<FONT color="green">022</FONT>    import java.util.Map;<a name="line.22"></a>
<FONT color="green">023</FONT>    <a name="line.23"></a>
<FONT color="green">024</FONT>    import org.apache.commons.math3.fraction.BigFraction;<a name="line.24"></a>
<FONT color="green">025</FONT>    import org.apache.commons.math3.util.ArithmeticUtils;<a name="line.25"></a>
<FONT color="green">026</FONT>    import org.apache.commons.math3.util.FastMath;<a name="line.26"></a>
<FONT color="green">027</FONT>    <a name="line.27"></a>
<FONT color="green">028</FONT>    /**<a name="line.28"></a>
<FONT color="green">029</FONT>     * A collection of static methods that operate on or return polynomials.<a name="line.29"></a>
<FONT color="green">030</FONT>     *<a name="line.30"></a>
<FONT color="green">031</FONT>     * @version $Id: PolynomialsUtils.java 1364387 2012-07-22 18:14:11Z tn $<a name="line.31"></a>
<FONT color="green">032</FONT>     * @since 2.0<a name="line.32"></a>
<FONT color="green">033</FONT>     */<a name="line.33"></a>
<FONT color="green">034</FONT>    public class PolynomialsUtils {<a name="line.34"></a>
<FONT color="green">035</FONT>    <a name="line.35"></a>
<FONT color="green">036</FONT>        /** Coefficients for Chebyshev polynomials. */<a name="line.36"></a>
<FONT color="green">037</FONT>        private static final List&lt;BigFraction&gt; CHEBYSHEV_COEFFICIENTS;<a name="line.37"></a>
<FONT color="green">038</FONT>    <a name="line.38"></a>
<FONT color="green">039</FONT>        /** Coefficients for Hermite polynomials. */<a name="line.39"></a>
<FONT color="green">040</FONT>        private static final List&lt;BigFraction&gt; HERMITE_COEFFICIENTS;<a name="line.40"></a>
<FONT color="green">041</FONT>    <a name="line.41"></a>
<FONT color="green">042</FONT>        /** Coefficients for Laguerre polynomials. */<a name="line.42"></a>
<FONT color="green">043</FONT>        private static final List&lt;BigFraction&gt; LAGUERRE_COEFFICIENTS;<a name="line.43"></a>
<FONT color="green">044</FONT>    <a name="line.44"></a>
<FONT color="green">045</FONT>        /** Coefficients for Legendre polynomials. */<a name="line.45"></a>
<FONT color="green">046</FONT>        private static final List&lt;BigFraction&gt; LEGENDRE_COEFFICIENTS;<a name="line.46"></a>
<FONT color="green">047</FONT>    <a name="line.47"></a>
<FONT color="green">048</FONT>        /** Coefficients for Jacobi polynomials. */<a name="line.48"></a>
<FONT color="green">049</FONT>        private static final Map&lt;JacobiKey, List&lt;BigFraction&gt;&gt; JACOBI_COEFFICIENTS;<a name="line.49"></a>
<FONT color="green">050</FONT>    <a name="line.50"></a>
<FONT color="green">051</FONT>        static {<a name="line.51"></a>
<FONT color="green">052</FONT>    <a name="line.52"></a>
<FONT color="green">053</FONT>            // initialize recurrence for Chebyshev polynomials<a name="line.53"></a>
<FONT color="green">054</FONT>            // T0(X) = 1, T1(X) = 0 + 1 * X<a name="line.54"></a>
<FONT color="green">055</FONT>            CHEBYSHEV_COEFFICIENTS = new ArrayList&lt;BigFraction&gt;();<a name="line.55"></a>
<FONT color="green">056</FONT>            CHEBYSHEV_COEFFICIENTS.add(BigFraction.ONE);<a name="line.56"></a>
<FONT color="green">057</FONT>            CHEBYSHEV_COEFFICIENTS.add(BigFraction.ZERO);<a name="line.57"></a>
<FONT color="green">058</FONT>            CHEBYSHEV_COEFFICIENTS.add(BigFraction.ONE);<a name="line.58"></a>
<FONT color="green">059</FONT>    <a name="line.59"></a>
<FONT color="green">060</FONT>            // initialize recurrence for Hermite polynomials<a name="line.60"></a>
<FONT color="green">061</FONT>            // H0(X) = 1, H1(X) = 0 + 2 * X<a name="line.61"></a>
<FONT color="green">062</FONT>            HERMITE_COEFFICIENTS = new ArrayList&lt;BigFraction&gt;();<a name="line.62"></a>
<FONT color="green">063</FONT>            HERMITE_COEFFICIENTS.add(BigFraction.ONE);<a name="line.63"></a>
<FONT color="green">064</FONT>            HERMITE_COEFFICIENTS.add(BigFraction.ZERO);<a name="line.64"></a>
<FONT color="green">065</FONT>            HERMITE_COEFFICIENTS.add(BigFraction.TWO);<a name="line.65"></a>
<FONT color="green">066</FONT>    <a name="line.66"></a>
<FONT color="green">067</FONT>            // initialize recurrence for Laguerre polynomials<a name="line.67"></a>
<FONT color="green">068</FONT>            // L0(X) = 1, L1(X) = 1 - 1 * X<a name="line.68"></a>
<FONT color="green">069</FONT>            LAGUERRE_COEFFICIENTS = new ArrayList&lt;BigFraction&gt;();<a name="line.69"></a>
<FONT color="green">070</FONT>            LAGUERRE_COEFFICIENTS.add(BigFraction.ONE);<a name="line.70"></a>
<FONT color="green">071</FONT>            LAGUERRE_COEFFICIENTS.add(BigFraction.ONE);<a name="line.71"></a>
<FONT color="green">072</FONT>            LAGUERRE_COEFFICIENTS.add(BigFraction.MINUS_ONE);<a name="line.72"></a>
<FONT color="green">073</FONT>    <a name="line.73"></a>
<FONT color="green">074</FONT>            // initialize recurrence for Legendre polynomials<a name="line.74"></a>
<FONT color="green">075</FONT>            // P0(X) = 1, P1(X) = 0 + 1 * X<a name="line.75"></a>
<FONT color="green">076</FONT>            LEGENDRE_COEFFICIENTS = new ArrayList&lt;BigFraction&gt;();<a name="line.76"></a>
<FONT color="green">077</FONT>            LEGENDRE_COEFFICIENTS.add(BigFraction.ONE);<a name="line.77"></a>
<FONT color="green">078</FONT>            LEGENDRE_COEFFICIENTS.add(BigFraction.ZERO);<a name="line.78"></a>
<FONT color="green">079</FONT>            LEGENDRE_COEFFICIENTS.add(BigFraction.ONE);<a name="line.79"></a>
<FONT color="green">080</FONT>    <a name="line.80"></a>
<FONT color="green">081</FONT>            // initialize map for Jacobi polynomials<a name="line.81"></a>
<FONT color="green">082</FONT>            JACOBI_COEFFICIENTS = new HashMap&lt;JacobiKey, List&lt;BigFraction&gt;&gt;();<a name="line.82"></a>
<FONT color="green">083</FONT>    <a name="line.83"></a>
<FONT color="green">084</FONT>        }<a name="line.84"></a>
<FONT color="green">085</FONT>    <a name="line.85"></a>
<FONT color="green">086</FONT>        /**<a name="line.86"></a>
<FONT color="green">087</FONT>         * Private constructor, to prevent instantiation.<a name="line.87"></a>
<FONT color="green">088</FONT>         */<a name="line.88"></a>
<FONT color="green">089</FONT>        private PolynomialsUtils() {<a name="line.89"></a>
<FONT color="green">090</FONT>        }<a name="line.90"></a>
<FONT color="green">091</FONT>    <a name="line.91"></a>
<FONT color="green">092</FONT>        /**<a name="line.92"></a>
<FONT color="green">093</FONT>         * Create a Chebyshev polynomial of the first kind.<a name="line.93"></a>
<FONT color="green">094</FONT>         * &lt;p&gt;&lt;a href="http://mathworld.wolfram.com/ChebyshevPolynomialoftheFirstKind.html"&gt;Chebyshev<a name="line.94"></a>
<FONT color="green">095</FONT>         * polynomials of the first kind&lt;/a&gt; are orthogonal polynomials.<a name="line.95"></a>
<FONT color="green">096</FONT>         * They can be defined by the following recurrence relations:<a name="line.96"></a>
<FONT color="green">097</FONT>         * &lt;pre&gt;<a name="line.97"></a>
<FONT color="green">098</FONT>         *  T&lt;sub&gt;0&lt;/sub&gt;(X)   = 1<a name="line.98"></a>
<FONT color="green">099</FONT>         *  T&lt;sub&gt;1&lt;/sub&gt;(X)   = X<a name="line.99"></a>
<FONT color="green">100</FONT>         *  T&lt;sub&gt;k+1&lt;/sub&gt;(X) = 2X T&lt;sub&gt;k&lt;/sub&gt;(X) - T&lt;sub&gt;k-1&lt;/sub&gt;(X)<a name="line.100"></a>
<FONT color="green">101</FONT>         * &lt;/pre&gt;&lt;/p&gt;<a name="line.101"></a>
<FONT color="green">102</FONT>         * @param degree degree of the polynomial<a name="line.102"></a>
<FONT color="green">103</FONT>         * @return Chebyshev polynomial of specified degree<a name="line.103"></a>
<FONT color="green">104</FONT>         */<a name="line.104"></a>
<FONT color="green">105</FONT>        public static PolynomialFunction createChebyshevPolynomial(final int degree) {<a name="line.105"></a>
<FONT color="green">106</FONT>            return buildPolynomial(degree, CHEBYSHEV_COEFFICIENTS,<a name="line.106"></a>
<FONT color="green">107</FONT>                    new RecurrenceCoefficientsGenerator() {<a name="line.107"></a>
<FONT color="green">108</FONT>                private final BigFraction[] coeffs = { BigFraction.ZERO, BigFraction.TWO, BigFraction.ONE };<a name="line.108"></a>
<FONT color="green">109</FONT>                /** {@inheritDoc} */<a name="line.109"></a>
<FONT color="green">110</FONT>                public BigFraction[] generate(int k) {<a name="line.110"></a>
<FONT color="green">111</FONT>                    return coeffs;<a name="line.111"></a>
<FONT color="green">112</FONT>                }<a name="line.112"></a>
<FONT color="green">113</FONT>            });<a name="line.113"></a>
<FONT color="green">114</FONT>        }<a name="line.114"></a>
<FONT color="green">115</FONT>    <a name="line.115"></a>
<FONT color="green">116</FONT>        /**<a name="line.116"></a>
<FONT color="green">117</FONT>         * Create a Hermite polynomial.<a name="line.117"></a>
<FONT color="green">118</FONT>         * &lt;p&gt;&lt;a href="http://mathworld.wolfram.com/HermitePolynomial.html"&gt;Hermite<a name="line.118"></a>
<FONT color="green">119</FONT>         * polynomials&lt;/a&gt; are orthogonal polynomials.<a name="line.119"></a>
<FONT color="green">120</FONT>         * They can be defined by the following recurrence relations:<a name="line.120"></a>
<FONT color="green">121</FONT>         * &lt;pre&gt;<a name="line.121"></a>
<FONT color="green">122</FONT>         *  H&lt;sub&gt;0&lt;/sub&gt;(X)   = 1<a name="line.122"></a>
<FONT color="green">123</FONT>         *  H&lt;sub&gt;1&lt;/sub&gt;(X)   = 2X<a name="line.123"></a>
<FONT color="green">124</FONT>         *  H&lt;sub&gt;k+1&lt;/sub&gt;(X) = 2X H&lt;sub&gt;k&lt;/sub&gt;(X) - 2k H&lt;sub&gt;k-1&lt;/sub&gt;(X)<a name="line.124"></a>
<FONT color="green">125</FONT>         * &lt;/pre&gt;&lt;/p&gt;<a name="line.125"></a>
<FONT color="green">126</FONT>    <a name="line.126"></a>
<FONT color="green">127</FONT>         * @param degree degree of the polynomial<a name="line.127"></a>
<FONT color="green">128</FONT>         * @return Hermite polynomial of specified degree<a name="line.128"></a>
<FONT color="green">129</FONT>         */<a name="line.129"></a>
<FONT color="green">130</FONT>        public static PolynomialFunction createHermitePolynomial(final int degree) {<a name="line.130"></a>
<FONT color="green">131</FONT>            return buildPolynomial(degree, HERMITE_COEFFICIENTS,<a name="line.131"></a>
<FONT color="green">132</FONT>                    new RecurrenceCoefficientsGenerator() {<a name="line.132"></a>
<FONT color="green">133</FONT>                /** {@inheritDoc} */<a name="line.133"></a>
<FONT color="green">134</FONT>                public BigFraction[] generate(int k) {<a name="line.134"></a>
<FONT color="green">135</FONT>                    return new BigFraction[] {<a name="line.135"></a>
<FONT color="green">136</FONT>                            BigFraction.ZERO,<a name="line.136"></a>
<FONT color="green">137</FONT>                            BigFraction.TWO,<a name="line.137"></a>
<FONT color="green">138</FONT>                            new BigFraction(2 * k)};<a name="line.138"></a>
<FONT color="green">139</FONT>                }<a name="line.139"></a>
<FONT color="green">140</FONT>            });<a name="line.140"></a>
<FONT color="green">141</FONT>        }<a name="line.141"></a>
<FONT color="green">142</FONT>    <a name="line.142"></a>
<FONT color="green">143</FONT>        /**<a name="line.143"></a>
<FONT color="green">144</FONT>         * Create a Laguerre polynomial.<a name="line.144"></a>
<FONT color="green">145</FONT>         * &lt;p&gt;&lt;a href="http://mathworld.wolfram.com/LaguerrePolynomial.html"&gt;Laguerre<a name="line.145"></a>
<FONT color="green">146</FONT>         * polynomials&lt;/a&gt; are orthogonal polynomials.<a name="line.146"></a>
<FONT color="green">147</FONT>         * They can be defined by the following recurrence relations:<a name="line.147"></a>
<FONT color="green">148</FONT>         * &lt;pre&gt;<a name="line.148"></a>
<FONT color="green">149</FONT>         *        L&lt;sub&gt;0&lt;/sub&gt;(X)   = 1<a name="line.149"></a>
<FONT color="green">150</FONT>         *        L&lt;sub&gt;1&lt;/sub&gt;(X)   = 1 - X<a name="line.150"></a>
<FONT color="green">151</FONT>         *  (k+1) L&lt;sub&gt;k+1&lt;/sub&gt;(X) = (2k + 1 - X) L&lt;sub&gt;k&lt;/sub&gt;(X) - k L&lt;sub&gt;k-1&lt;/sub&gt;(X)<a name="line.151"></a>
<FONT color="green">152</FONT>         * &lt;/pre&gt;&lt;/p&gt;<a name="line.152"></a>
<FONT color="green">153</FONT>         * @param degree degree of the polynomial<a name="line.153"></a>
<FONT color="green">154</FONT>         * @return Laguerre polynomial of specified degree<a name="line.154"></a>
<FONT color="green">155</FONT>         */<a name="line.155"></a>
<FONT color="green">156</FONT>        public static PolynomialFunction createLaguerrePolynomial(final int degree) {<a name="line.156"></a>
<FONT color="green">157</FONT>            return buildPolynomial(degree, LAGUERRE_COEFFICIENTS,<a name="line.157"></a>
<FONT color="green">158</FONT>                    new RecurrenceCoefficientsGenerator() {<a name="line.158"></a>
<FONT color="green">159</FONT>                /** {@inheritDoc} */<a name="line.159"></a>
<FONT color="green">160</FONT>                public BigFraction[] generate(int k) {<a name="line.160"></a>
<FONT color="green">161</FONT>                    final int kP1 = k + 1;<a name="line.161"></a>
<FONT color="green">162</FONT>                    return new BigFraction[] {<a name="line.162"></a>
<FONT color="green">163</FONT>                            new BigFraction(2 * k + 1, kP1),<a name="line.163"></a>
<FONT color="green">164</FONT>                            new BigFraction(-1, kP1),<a name="line.164"></a>
<FONT color="green">165</FONT>                            new BigFraction(k, kP1)};<a name="line.165"></a>
<FONT color="green">166</FONT>                }<a name="line.166"></a>
<FONT color="green">167</FONT>            });<a name="line.167"></a>
<FONT color="green">168</FONT>        }<a name="line.168"></a>
<FONT color="green">169</FONT>    <a name="line.169"></a>
<FONT color="green">170</FONT>        /**<a name="line.170"></a>
<FONT color="green">171</FONT>         * Create a Legendre polynomial.<a name="line.171"></a>
<FONT color="green">172</FONT>         * &lt;p&gt;&lt;a href="http://mathworld.wolfram.com/LegendrePolynomial.html"&gt;Legendre<a name="line.172"></a>
<FONT color="green">173</FONT>         * polynomials&lt;/a&gt; are orthogonal polynomials.<a name="line.173"></a>
<FONT color="green">174</FONT>         * They can be defined by the following recurrence relations:<a name="line.174"></a>
<FONT color="green">175</FONT>         * &lt;pre&gt;<a name="line.175"></a>
<FONT color="green">176</FONT>         *        P&lt;sub&gt;0&lt;/sub&gt;(X)   = 1<a name="line.176"></a>
<FONT color="green">177</FONT>         *        P&lt;sub&gt;1&lt;/sub&gt;(X)   = X<a name="line.177"></a>
<FONT color="green">178</FONT>         *  (k+1) P&lt;sub&gt;k+1&lt;/sub&gt;(X) = (2k+1) X P&lt;sub&gt;k&lt;/sub&gt;(X) - k P&lt;sub&gt;k-1&lt;/sub&gt;(X)<a name="line.178"></a>
<FONT color="green">179</FONT>         * &lt;/pre&gt;&lt;/p&gt;<a name="line.179"></a>
<FONT color="green">180</FONT>         * @param degree degree of the polynomial<a name="line.180"></a>
<FONT color="green">181</FONT>         * @return Legendre polynomial of specified degree<a name="line.181"></a>
<FONT color="green">182</FONT>         */<a name="line.182"></a>
<FONT color="green">183</FONT>        public static PolynomialFunction createLegendrePolynomial(final int degree) {<a name="line.183"></a>
<FONT color="green">184</FONT>            return buildPolynomial(degree, LEGENDRE_COEFFICIENTS,<a name="line.184"></a>
<FONT color="green">185</FONT>                                   new RecurrenceCoefficientsGenerator() {<a name="line.185"></a>
<FONT color="green">186</FONT>                /** {@inheritDoc} */<a name="line.186"></a>
<FONT color="green">187</FONT>                public BigFraction[] generate(int k) {<a name="line.187"></a>
<FONT color="green">188</FONT>                    final int kP1 = k + 1;<a name="line.188"></a>
<FONT color="green">189</FONT>                    return new BigFraction[] {<a name="line.189"></a>
<FONT color="green">190</FONT>                            BigFraction.ZERO,<a name="line.190"></a>
<FONT color="green">191</FONT>                            new BigFraction(k + kP1, kP1),<a name="line.191"></a>
<FONT color="green">192</FONT>                            new BigFraction(k, kP1)};<a name="line.192"></a>
<FONT color="green">193</FONT>                }<a name="line.193"></a>
<FONT color="green">194</FONT>            });<a name="line.194"></a>
<FONT color="green">195</FONT>        }<a name="line.195"></a>
<FONT color="green">196</FONT>    <a name="line.196"></a>
<FONT color="green">197</FONT>        /**<a name="line.197"></a>
<FONT color="green">198</FONT>         * Create a Jacobi polynomial.<a name="line.198"></a>
<FONT color="green">199</FONT>         * &lt;p&gt;&lt;a href="http://mathworld.wolfram.com/JacobiPolynomial.html"&gt;Jacobi<a name="line.199"></a>
<FONT color="green">200</FONT>         * polynomials&lt;/a&gt; are orthogonal polynomials.<a name="line.200"></a>
<FONT color="green">201</FONT>         * They can be defined by the following recurrence relations:<a name="line.201"></a>
<FONT color="green">202</FONT>         * &lt;pre&gt;<a name="line.202"></a>
<FONT color="green">203</FONT>         *        P&lt;sub&gt;0&lt;/sub&gt;&lt;sup&gt;vw&lt;/sup&gt;(X)   = 1<a name="line.203"></a>
<FONT color="green">204</FONT>         *        P&lt;sub&gt;-1&lt;/sub&gt;&lt;sup&gt;vw&lt;/sup&gt;(X)  = 0<a name="line.204"></a>
<FONT color="green">205</FONT>         *  2k(k + v + w)(2k + v + w - 2) P&lt;sub&gt;k&lt;/sub&gt;&lt;sup&gt;vw&lt;/sup&gt;(X) =<a name="line.205"></a>
<FONT color="green">206</FONT>         *  (2k + v + w - 1)[(2k + v + w)(2k + v + w - 2) X + v&lt;sup&gt;2&lt;/sup&gt; - w&lt;sup&gt;2&lt;/sup&gt;] P&lt;sub&gt;k-1&lt;/sub&gt;&lt;sup&gt;vw&lt;/sup&gt;(X)<a name="line.206"></a>
<FONT color="green">207</FONT>         *  - 2(k + v - 1)(k + w - 1)(2k + v + w) P&lt;sub&gt;k-2&lt;/sub&gt;&lt;sup&gt;vw&lt;/sup&gt;(X)<a name="line.207"></a>
<FONT color="green">208</FONT>         * &lt;/pre&gt;&lt;/p&gt;<a name="line.208"></a>
<FONT color="green">209</FONT>         * @param degree degree of the polynomial<a name="line.209"></a>
<FONT color="green">210</FONT>         * @param v first exponent<a name="line.210"></a>
<FONT color="green">211</FONT>         * @param w second exponent<a name="line.211"></a>
<FONT color="green">212</FONT>         * @return Jacobi polynomial of specified degree<a name="line.212"></a>
<FONT color="green">213</FONT>         */<a name="line.213"></a>
<FONT color="green">214</FONT>        public static PolynomialFunction createJacobiPolynomial(final int degree, final int v, final int w) {<a name="line.214"></a>
<FONT color="green">215</FONT>    <a name="line.215"></a>
<FONT color="green">216</FONT>            // select the appropriate list<a name="line.216"></a>
<FONT color="green">217</FONT>            final JacobiKey key = new JacobiKey(v, w);<a name="line.217"></a>
<FONT color="green">218</FONT>    <a name="line.218"></a>
<FONT color="green">219</FONT>            if (!JACOBI_COEFFICIENTS.containsKey(key)) {<a name="line.219"></a>
<FONT color="green">220</FONT>    <a name="line.220"></a>
<FONT color="green">221</FONT>                // allocate a new list for v, w<a name="line.221"></a>
<FONT color="green">222</FONT>                final List&lt;BigFraction&gt; list = new ArrayList&lt;BigFraction&gt;();<a name="line.222"></a>
<FONT color="green">223</FONT>                JACOBI_COEFFICIENTS.put(key, list);<a name="line.223"></a>
<FONT color="green">224</FONT>    <a name="line.224"></a>
<FONT color="green">225</FONT>                // Pv,w,0(x) = 1;<a name="line.225"></a>
<FONT color="green">226</FONT>                list.add(BigFraction.ONE);<a name="line.226"></a>
<FONT color="green">227</FONT>    <a name="line.227"></a>
<FONT color="green">228</FONT>                // P1(x) = (v - w) / 2 + (2 + v + w) * X / 2<a name="line.228"></a>
<FONT color="green">229</FONT>                list.add(new BigFraction(v - w, 2));<a name="line.229"></a>
<FONT color="green">230</FONT>                list.add(new BigFraction(2 + v + w, 2));<a name="line.230"></a>
<FONT color="green">231</FONT>    <a name="line.231"></a>
<FONT color="green">232</FONT>            }<a name="line.232"></a>
<FONT color="green">233</FONT>    <a name="line.233"></a>
<FONT color="green">234</FONT>            return buildPolynomial(degree, JACOBI_COEFFICIENTS.get(key),<a name="line.234"></a>
<FONT color="green">235</FONT>                                   new RecurrenceCoefficientsGenerator() {<a name="line.235"></a>
<FONT color="green">236</FONT>                /** {@inheritDoc} */<a name="line.236"></a>
<FONT color="green">237</FONT>                public BigFraction[] generate(int k) {<a name="line.237"></a>
<FONT color="green">238</FONT>                    k++;<a name="line.238"></a>
<FONT color="green">239</FONT>                    final int kvw      = k + v + w;<a name="line.239"></a>
<FONT color="green">240</FONT>                    final int twoKvw   = kvw + k;<a name="line.240"></a>
<FONT color="green">241</FONT>                    final int twoKvwM1 = twoKvw - 1;<a name="line.241"></a>
<FONT color="green">242</FONT>                    final int twoKvwM2 = twoKvw - 2;<a name="line.242"></a>
<FONT color="green">243</FONT>                    final int den      = 2 * k *  kvw * twoKvwM2;<a name="line.243"></a>
<FONT color="green">244</FONT>    <a name="line.244"></a>
<FONT color="green">245</FONT>                    return new BigFraction[] {<a name="line.245"></a>
<FONT color="green">246</FONT>                        new BigFraction(twoKvwM1 * (v * v - w * w), den),<a name="line.246"></a>
<FONT color="green">247</FONT>                        new BigFraction(twoKvwM1 * twoKvw * twoKvwM2, den),<a name="line.247"></a>
<FONT color="green">248</FONT>                        new BigFraction(2 * (k + v - 1) * (k + w - 1) * twoKvw, den)<a name="line.248"></a>
<FONT color="green">249</FONT>                    };<a name="line.249"></a>
<FONT color="green">250</FONT>                }<a name="line.250"></a>
<FONT color="green">251</FONT>            });<a name="line.251"></a>
<FONT color="green">252</FONT>    <a name="line.252"></a>
<FONT color="green">253</FONT>        }<a name="line.253"></a>
<FONT color="green">254</FONT>    <a name="line.254"></a>
<FONT color="green">255</FONT>        /** Inner class for Jacobi polynomials keys. */<a name="line.255"></a>
<FONT color="green">256</FONT>        private static class JacobiKey {<a name="line.256"></a>
<FONT color="green">257</FONT>    <a name="line.257"></a>
<FONT color="green">258</FONT>            /** First exponent. */<a name="line.258"></a>
<FONT color="green">259</FONT>            private final int v;<a name="line.259"></a>
<FONT color="green">260</FONT>    <a name="line.260"></a>
<FONT color="green">261</FONT>            /** Second exponent. */<a name="line.261"></a>
<FONT color="green">262</FONT>            private final int w;<a name="line.262"></a>
<FONT color="green">263</FONT>    <a name="line.263"></a>
<FONT color="green">264</FONT>            /** Simple constructor.<a name="line.264"></a>
<FONT color="green">265</FONT>             * @param v first exponent<a name="line.265"></a>
<FONT color="green">266</FONT>             * @param w second exponent<a name="line.266"></a>
<FONT color="green">267</FONT>             */<a name="line.267"></a>
<FONT color="green">268</FONT>            public JacobiKey(final int v, final int w) {<a name="line.268"></a>
<FONT color="green">269</FONT>                this.v = v;<a name="line.269"></a>
<FONT color="green">270</FONT>                this.w = w;<a name="line.270"></a>
<FONT color="green">271</FONT>            }<a name="line.271"></a>
<FONT color="green">272</FONT>    <a name="line.272"></a>
<FONT color="green">273</FONT>            /** Get hash code.<a name="line.273"></a>
<FONT color="green">274</FONT>             * @return hash code<a name="line.274"></a>
<FONT color="green">275</FONT>             */<a name="line.275"></a>
<FONT color="green">276</FONT>            @Override<a name="line.276"></a>
<FONT color="green">277</FONT>            public int hashCode() {<a name="line.277"></a>
<FONT color="green">278</FONT>                return (v &lt;&lt; 16) ^ w;<a name="line.278"></a>
<FONT color="green">279</FONT>            }<a name="line.279"></a>
<FONT color="green">280</FONT>    <a name="line.280"></a>
<FONT color="green">281</FONT>            /** Check if the instance represent the same key as another instance.<a name="line.281"></a>
<FONT color="green">282</FONT>             * @param key other key<a name="line.282"></a>
<FONT color="green">283</FONT>             * @return true if the instance and the other key refer to the same polynomial<a name="line.283"></a>
<FONT color="green">284</FONT>             */<a name="line.284"></a>
<FONT color="green">285</FONT>            @Override<a name="line.285"></a>
<FONT color="green">286</FONT>            public boolean equals(final Object key) {<a name="line.286"></a>
<FONT color="green">287</FONT>    <a name="line.287"></a>
<FONT color="green">288</FONT>                if ((key == null) || !(key instanceof JacobiKey)) {<a name="line.288"></a>
<FONT color="green">289</FONT>                    return false;<a name="line.289"></a>
<FONT color="green">290</FONT>                }<a name="line.290"></a>
<FONT color="green">291</FONT>    <a name="line.291"></a>
<FONT color="green">292</FONT>                final JacobiKey otherK = (JacobiKey) key;<a name="line.292"></a>
<FONT color="green">293</FONT>                return (v == otherK.v) &amp;&amp; (w == otherK.w);<a name="line.293"></a>
<FONT color="green">294</FONT>    <a name="line.294"></a>
<FONT color="green">295</FONT>            }<a name="line.295"></a>
<FONT color="green">296</FONT>        }<a name="line.296"></a>
<FONT color="green">297</FONT>    <a name="line.297"></a>
<FONT color="green">298</FONT>        /**<a name="line.298"></a>
<FONT color="green">299</FONT>         * Compute the coefficients of the polynomial &lt;code&gt;P&lt;sub&gt;s&lt;/sub&gt;(x)&lt;/code&gt;<a name="line.299"></a>
<FONT color="green">300</FONT>         * whose values at point {@code x} will be the same as the those from the<a name="line.300"></a>
<FONT color="green">301</FONT>         * original polynomial &lt;code&gt;P(x)&lt;/code&gt; when computed at {@code x + shift}.<a name="line.301"></a>
<FONT color="green">302</FONT>         * Thus, if &lt;code&gt;P(x) = &amp;Sigma;&lt;sub&gt;i&lt;/sub&gt; a&lt;sub&gt;i&lt;/sub&gt; x&lt;sup&gt;i&lt;/sup&gt;&lt;/code&gt;,<a name="line.302"></a>
<FONT color="green">303</FONT>         * then<a name="line.303"></a>
<FONT color="green">304</FONT>         * &lt;pre&gt;<a name="line.304"></a>
<FONT color="green">305</FONT>         *  &lt;table&gt;<a name="line.305"></a>
<FONT color="green">306</FONT>         *   &lt;tr&gt;<a name="line.306"></a>
<FONT color="green">307</FONT>         *    &lt;td&gt;&lt;code&gt;P&lt;sub&gt;s&lt;/sub&gt;(x)&lt;/td&gt;<a name="line.307"></a>
<FONT color="green">308</FONT>         *    &lt;td&gt;= &amp;Sigma;&lt;sub&gt;i&lt;/sub&gt; b&lt;sub&gt;i&lt;/sub&gt; x&lt;sup&gt;i&lt;/sup&gt;&lt;/code&gt;&lt;/td&gt;<a name="line.308"></a>
<FONT color="green">309</FONT>         *   &lt;/tr&gt;<a name="line.309"></a>
<FONT color="green">310</FONT>         *   &lt;tr&gt;<a name="line.310"></a>
<FONT color="green">311</FONT>         *    &lt;td&gt;&lt;/td&gt;<a name="line.311"></a>
<FONT color="green">312</FONT>         *    &lt;td&gt;= &amp;Sigma;&lt;sub&gt;i&lt;/sub&gt; a&lt;sub&gt;i&lt;/sub&gt; (x + shift)&lt;sup&gt;i&lt;/sup&gt;&lt;/code&gt;&lt;/td&gt;<a name="line.312"></a>
<FONT color="green">313</FONT>         *   &lt;/tr&gt;<a name="line.313"></a>
<FONT color="green">314</FONT>         *  &lt;/table&gt;<a name="line.314"></a>
<FONT color="green">315</FONT>         * &lt;/pre&gt;<a name="line.315"></a>
<FONT color="green">316</FONT>         *<a name="line.316"></a>
<FONT color="green">317</FONT>         * @param coefficients Coefficients of the original polynomial.<a name="line.317"></a>
<FONT color="green">318</FONT>         * @param shift Shift value.<a name="line.318"></a>
<FONT color="green">319</FONT>         * @return the coefficients &lt;code&gt;b&lt;sub&gt;i&lt;/sub&gt;&lt;/code&gt; of the shifted<a name="line.319"></a>
<FONT color="green">320</FONT>         * polynomial.<a name="line.320"></a>
<FONT color="green">321</FONT>         */<a name="line.321"></a>
<FONT color="green">322</FONT>        public static double[] shift(final double[] coefficients,<a name="line.322"></a>
<FONT color="green">323</FONT>                                     final double shift) {<a name="line.323"></a>
<FONT color="green">324</FONT>            final int dp1 = coefficients.length;<a name="line.324"></a>
<FONT color="green">325</FONT>            final double[] newCoefficients = new double[dp1];<a name="line.325"></a>
<FONT color="green">326</FONT>    <a name="line.326"></a>
<FONT color="green">327</FONT>            // Pascal triangle.<a name="line.327"></a>
<FONT color="green">328</FONT>            final int[][] coeff = new int[dp1][dp1];<a name="line.328"></a>
<FONT color="green">329</FONT>            for (int i = 0; i &lt; dp1; i++){<a name="line.329"></a>
<FONT color="green">330</FONT>                for(int j = 0; j &lt;= i; j++){<a name="line.330"></a>
<FONT color="green">331</FONT>                    coeff[i][j] = (int) ArithmeticUtils.binomialCoefficient(i, j);<a name="line.331"></a>
<FONT color="green">332</FONT>                }<a name="line.332"></a>
<FONT color="green">333</FONT>            }<a name="line.333"></a>
<FONT color="green">334</FONT>    <a name="line.334"></a>
<FONT color="green">335</FONT>            // First polynomial coefficient.<a name="line.335"></a>
<FONT color="green">336</FONT>            for (int i = 0; i &lt; dp1; i++){<a name="line.336"></a>
<FONT color="green">337</FONT>                newCoefficients[0] += coefficients[i] * FastMath.pow(shift, i);<a name="line.337"></a>
<FONT color="green">338</FONT>            }<a name="line.338"></a>
<FONT color="green">339</FONT>    <a name="line.339"></a>
<FONT color="green">340</FONT>            // Superior order.<a name="line.340"></a>
<FONT color="green">341</FONT>            final int d = dp1 - 1;<a name="line.341"></a>
<FONT color="green">342</FONT>            for (int i = 0; i &lt; d; i++) {<a name="line.342"></a>
<FONT color="green">343</FONT>                for (int j = i; j &lt; d; j++){<a name="line.343"></a>
<FONT color="green">344</FONT>                    newCoefficients[i + 1] += coeff[j + 1][j - i] *<a name="line.344"></a>
<FONT color="green">345</FONT>                        coefficients[j + 1] * FastMath.pow(shift, j - i);<a name="line.345"></a>
<FONT color="green">346</FONT>                }<a name="line.346"></a>
<FONT color="green">347</FONT>            }<a name="line.347"></a>
<FONT color="green">348</FONT>    <a name="line.348"></a>
<FONT color="green">349</FONT>            return newCoefficients;<a name="line.349"></a>
<FONT color="green">350</FONT>        }<a name="line.350"></a>
<FONT color="green">351</FONT>    <a name="line.351"></a>
<FONT color="green">352</FONT>    <a name="line.352"></a>
<FONT color="green">353</FONT>        /** Get the coefficients array for a given degree.<a name="line.353"></a>
<FONT color="green">354</FONT>         * @param degree degree of the polynomial<a name="line.354"></a>
<FONT color="green">355</FONT>         * @param coefficients list where the computed coefficients are stored<a name="line.355"></a>
<FONT color="green">356</FONT>         * @param generator recurrence coefficients generator<a name="line.356"></a>
<FONT color="green">357</FONT>         * @return coefficients array<a name="line.357"></a>
<FONT color="green">358</FONT>         */<a name="line.358"></a>
<FONT color="green">359</FONT>        private static PolynomialFunction buildPolynomial(final int degree,<a name="line.359"></a>
<FONT color="green">360</FONT>                                                          final List&lt;BigFraction&gt; coefficients,<a name="line.360"></a>
<FONT color="green">361</FONT>                                                          final RecurrenceCoefficientsGenerator generator) {<a name="line.361"></a>
<FONT color="green">362</FONT>    <a name="line.362"></a>
<FONT color="green">363</FONT>            final int maxDegree = (int) FastMath.floor(FastMath.sqrt(2 * coefficients.size())) - 1;<a name="line.363"></a>
<FONT color="green">364</FONT>            synchronized (PolynomialsUtils.class) {<a name="line.364"></a>
<FONT color="green">365</FONT>                if (degree &gt; maxDegree) {<a name="line.365"></a>
<FONT color="green">366</FONT>                    computeUpToDegree(degree, maxDegree, generator, coefficients);<a name="line.366"></a>
<FONT color="green">367</FONT>                }<a name="line.367"></a>
<FONT color="green">368</FONT>            }<a name="line.368"></a>
<FONT color="green">369</FONT>    <a name="line.369"></a>
<FONT color="green">370</FONT>            // coefficient  for polynomial 0 is  l [0]<a name="line.370"></a>
<FONT color="green">371</FONT>            // coefficients for polynomial 1 are l [1] ... l [2] (degrees 0 ... 1)<a name="line.371"></a>
<FONT color="green">372</FONT>            // coefficients for polynomial 2 are l [3] ... l [5] (degrees 0 ... 2)<a name="line.372"></a>
<FONT color="green">373</FONT>            // coefficients for polynomial 3 are l [6] ... l [9] (degrees 0 ... 3)<a name="line.373"></a>
<FONT color="green">374</FONT>            // coefficients for polynomial 4 are l[10] ... l[14] (degrees 0 ... 4)<a name="line.374"></a>
<FONT color="green">375</FONT>            // coefficients for polynomial 5 are l[15] ... l[20] (degrees 0 ... 5)<a name="line.375"></a>
<FONT color="green">376</FONT>            // coefficients for polynomial 6 are l[21] ... l[27] (degrees 0 ... 6)<a name="line.376"></a>
<FONT color="green">377</FONT>            // ...<a name="line.377"></a>
<FONT color="green">378</FONT>            final int start = degree * (degree + 1) / 2;<a name="line.378"></a>
<FONT color="green">379</FONT>    <a name="line.379"></a>
<FONT color="green">380</FONT>            final double[] a = new double[degree + 1];<a name="line.380"></a>
<FONT color="green">381</FONT>            for (int i = 0; i &lt;= degree; ++i) {<a name="line.381"></a>
<FONT color="green">382</FONT>                a[i] = coefficients.get(start + i).doubleValue();<a name="line.382"></a>
<FONT color="green">383</FONT>            }<a name="line.383"></a>
<FONT color="green">384</FONT>    <a name="line.384"></a>
<FONT color="green">385</FONT>            // build the polynomial<a name="line.385"></a>
<FONT color="green">386</FONT>            return new PolynomialFunction(a);<a name="line.386"></a>
<FONT color="green">387</FONT>    <a name="line.387"></a>
<FONT color="green">388</FONT>        }<a name="line.388"></a>
<FONT color="green">389</FONT>    <a name="line.389"></a>
<FONT color="green">390</FONT>        /** Compute polynomial coefficients up to a given degree.<a name="line.390"></a>
<FONT color="green">391</FONT>         * @param degree maximal degree<a name="line.391"></a>
<FONT color="green">392</FONT>         * @param maxDegree current maximal degree<a name="line.392"></a>
<FONT color="green">393</FONT>         * @param generator recurrence coefficients generator<a name="line.393"></a>
<FONT color="green">394</FONT>         * @param coefficients list where the computed coefficients should be appended<a name="line.394"></a>
<FONT color="green">395</FONT>         */<a name="line.395"></a>
<FONT color="green">396</FONT>        private static void computeUpToDegree(final int degree, final int maxDegree,<a name="line.396"></a>
<FONT color="green">397</FONT>                                              final RecurrenceCoefficientsGenerator generator,<a name="line.397"></a>
<FONT color="green">398</FONT>                                              final List&lt;BigFraction&gt; coefficients) {<a name="line.398"></a>
<FONT color="green">399</FONT>    <a name="line.399"></a>
<FONT color="green">400</FONT>            int startK = (maxDegree - 1) * maxDegree / 2;<a name="line.400"></a>
<FONT color="green">401</FONT>            for (int k = maxDegree; k &lt; degree; ++k) {<a name="line.401"></a>
<FONT color="green">402</FONT>    <a name="line.402"></a>
<FONT color="green">403</FONT>                // start indices of two previous polynomials Pk(X) and Pk-1(X)<a name="line.403"></a>
<FONT color="green">404</FONT>                int startKm1 = startK;<a name="line.404"></a>
<FONT color="green">405</FONT>                startK += k;<a name="line.405"></a>
<FONT color="green">406</FONT>    <a name="line.406"></a>
<FONT color="green">407</FONT>                // Pk+1(X) = (a[0] + a[1] X) Pk(X) - a[2] Pk-1(X)<a name="line.407"></a>
<FONT color="green">408</FONT>                BigFraction[] ai = generator.generate(k);<a name="line.408"></a>
<FONT color="green">409</FONT>    <a name="line.409"></a>
<FONT color="green">410</FONT>                BigFraction ck     = coefficients.get(startK);<a name="line.410"></a>
<FONT color="green">411</FONT>                BigFraction ckm1   = coefficients.get(startKm1);<a name="line.411"></a>
<FONT color="green">412</FONT>    <a name="line.412"></a>
<FONT color="green">413</FONT>                // degree 0 coefficient<a name="line.413"></a>
<FONT color="green">414</FONT>                coefficients.add(ck.multiply(ai[0]).subtract(ckm1.multiply(ai[2])));<a name="line.414"></a>
<FONT color="green">415</FONT>    <a name="line.415"></a>
<FONT color="green">416</FONT>                // degree 1 to degree k-1 coefficients<a name="line.416"></a>
<FONT color="green">417</FONT>                for (int i = 1; i &lt; k; ++i) {<a name="line.417"></a>
<FONT color="green">418</FONT>                    final BigFraction ckPrev = ck;<a name="line.418"></a>
<FONT color="green">419</FONT>                    ck     = coefficients.get(startK + i);<a name="line.419"></a>
<FONT color="green">420</FONT>                    ckm1   = coefficients.get(startKm1 + i);<a name="line.420"></a>
<FONT color="green">421</FONT>                    coefficients.add(ck.multiply(ai[0]).add(ckPrev.multiply(ai[1])).subtract(ckm1.multiply(ai[2])));<a name="line.421"></a>
<FONT color="green">422</FONT>                }<a name="line.422"></a>
<FONT color="green">423</FONT>    <a name="line.423"></a>
<FONT color="green">424</FONT>                // degree k coefficient<a name="line.424"></a>
<FONT color="green">425</FONT>                final BigFraction ckPrev = ck;<a name="line.425"></a>
<FONT color="green">426</FONT>                ck = coefficients.get(startK + k);<a name="line.426"></a>
<FONT color="green">427</FONT>                coefficients.add(ck.multiply(ai[0]).add(ckPrev.multiply(ai[1])));<a name="line.427"></a>
<FONT color="green">428</FONT>    <a name="line.428"></a>
<FONT color="green">429</FONT>                // degree k+1 coefficient<a name="line.429"></a>
<FONT color="green">430</FONT>                coefficients.add(ck.multiply(ai[1]));<a name="line.430"></a>
<FONT color="green">431</FONT>    <a name="line.431"></a>
<FONT color="green">432</FONT>            }<a name="line.432"></a>
<FONT color="green">433</FONT>    <a name="line.433"></a>
<FONT color="green">434</FONT>        }<a name="line.434"></a>
<FONT color="green">435</FONT>    <a name="line.435"></a>
<FONT color="green">436</FONT>        /** Interface for recurrence coefficients generation. */<a name="line.436"></a>
<FONT color="green">437</FONT>        private interface RecurrenceCoefficientsGenerator {<a name="line.437"></a>
<FONT color="green">438</FONT>            /**<a name="line.438"></a>
<FONT color="green">439</FONT>             * Generate recurrence coefficients.<a name="line.439"></a>
<FONT color="green">440</FONT>             * @param k highest degree of the polynomials used in the recurrence<a name="line.440"></a>
<FONT color="green">441</FONT>             * @return an array of three coefficients such that<a name="line.441"></a>
<FONT color="green">442</FONT>             * P&lt;sub&gt;k+1&lt;/sub&gt;(X) = (a[0] + a[1] X) P&lt;sub&gt;k&lt;/sub&gt;(X) - a[2] P&lt;sub&gt;k-1&lt;/sub&gt;(X)<a name="line.442"></a>
<FONT color="green">443</FONT>             */<a name="line.443"></a>
<FONT color="green">444</FONT>            BigFraction[] generate(int k);<a name="line.444"></a>
<FONT color="green">445</FONT>        }<a name="line.445"></a>
<FONT color="green">446</FONT>    <a name="line.446"></a>
<FONT color="green">447</FONT>    }<a name="line.447"></a>




























































</PRE>
</BODY>
</HTML>
